Figure 1

Simulation geometry: the layered media are distinguished by different shades and denoted by ${ϵ}_{1r}$, ${ϵ}_{2r}$, etc. Thick, dashed (thin, dotted) lines denote the boundaries to which the $H-\left(E-\right)$ field is assigned. The left, lower left, and lower right panels show the specific field point assignment at the interface, along line $a$, and at the horizontal boundaries, respectively. The $x$ coordinates of lines $a$, $b^{\prime }$, $b$, $c$, $c^{\prime }$, and $d$ are $i_1-1/2$, $i_1$, $i_1+1/2$, $i_2-1/2$, $i_2$, and $i_2+1/2$, respectively. The $y$ coordinates of lines $e$, $f$, $g$, and $h$ are $j_1$, $j_1+1/2$, $j_2-1/2$, and $j_2$, respectively. The perfectly matched layers absorbing boundaries are not shown.Reuse & Permissions

Figure 2

Simulation flow chart. Note that “IN” (OUT) refers to the input (output) side of the incident (final transmitted) wave. The input (output) side is at a smaller (larger) $y$ value in Fig. 1. For 1D field updates an ADE approach is used whereas 2D field updates are performed by either the ADE approach or the equations consistent with the UPML formulation.Reuse & Permissions

Figure 3

(Color online) In panels (a)–(c), the incident plane wave enters the TF region from the lower left corner of the TF/SF boundary with incident angle $\theta =65°$. (a) Magnetic $\left(H\right)$ field snapshot at 3.00 fs for a plane wave propagating in vacuum. The dashed oval indicates the leakage outside the TF/SF boundary as a result of the instantaneous turn-on of the field. Panels (b)–(d) show $H$-field snapshots for a plane wave propagation with initial Gaussian ramping. (b) $H$-field snapshot during ramping (at 7.34 fs) (c) $H$-field snapshot after steady state is established (at 66.71 fs), and (d) $H$-field snapshot at 90.06 fs for a plane wave propagating in the positive $y$ direction. For all calculations, the incident wavelength is $\lambda =400\text{nm}$ (period $T=1.33\text{fs}$), and the steady-state amplitude of the incident magnetic field is 1 A/m. The mesh size is $\Delta x=2.5\text{nm}$, and the Courant number is $S=0.4$. A log color scale $\left({\text{log}}_{10}\left|H_z\right|\right)$ is used, and the thick, dashed (thin, dotted) rectangle indicates the TF/SF (inner PML) boundary.Reuse & Permissions

Figure 4

(Color online) Magnetic field snapshots of a plane wave obliquely incident on a dielectric slab (indicated by a solid rectangle). Reflection and refraction (a) at the lower interface (at 10.01 fs) and (b) after steady state is established (at 100.07 fs). The incident plane wave enters the TF region from the lower left corner of the TF/SF boundary with incident angle $\theta =45°$. The incident wavelength in vacuum is $\lambda =400\text{nm}$ (period $T=1.33\text{fs}$) and the steady-state amplitude of the incident magnetic field is 1 A/m in all calculations. The dielectric constant and thickness of the slab are ${ϵ}_r=11.7\text{nm}$ and 900 nm, respectively. The media above and below the slab are vacuum, the mesh size is $\Delta x=1.0\text{nm}$, and the Courant number is $S=0.3$. A log color scale is used in all plots and the thick, dashed (thin, dotted) rectangle indicates the TF/SF (inner PML) boundary. The slab does not penetrate into the PML region.Reuse & Permissions

Figure 5

(Color online) Relative error in the magnitude of (a) the reflection coefficient $r$ and (b) the transmission coefficient $t$ as a function of the mesh size $\Delta x$. (c) Maximum leakage as a function of mesh size $\Delta x$. In all calculations, the Courant number is $S=0.3$. In all figures, the red, solid (blue, dashed) curve shows the result without (with) the interface averaging of the dielectric constants. The parameters of the incident wave and the dielectric slab are as in the calculation leading to Fig. 4.Reuse & Permissions

Figure 6

(Color online) Magnetic field snapshots of a plane wave obliquely incident on a Drude metal slab (indicated by a solid rectangle). Reflection, refraction and transmission (a) before (at 2.50 fs) and (b) after (at 60.04 fs) steady state is established. In all calculations, the incident plane wave enters the TF region from the lower left corner of the TF/SF boundary (dashed lines) with incident angle $\theta =45°$. The incident wavelength in vacuum is $\lambda =400\text{nm}$ (period $T=1.33\text{fs}$) and the steady-state amplitude of the incident magnetic field is 1 A/m. The metal slab is 80 nm thick with Drude parameters: $ϵ\left(\infty \right)=7.0246$, ${\omega }_D=1.5713\times {10}^{16}\text{rad}/\text{s}$, and ${\Gamma }_D=1.4003\times {10}^{14}\text{rad}/\text{s}$. The media above and below the slab are vacuum, the mesh size is $\Delta x=1.0\text{nm}$, and the Courant number is $S=0.3$. A log color scale is used in all plots and the thick, dashed (thin, dotted) rectangle indicates the TF/SF (inner PML) boundary. The slab does not penetrate into the PML region.Reuse & Permissions

Figure 7

(Color online) Relative error in the magnitude of (a) the reflection coefficient $r$ and (b) the transmission coefficient $t$ as a function of the mesh size $\Delta x$. (c) Maximum leakage as a function of mesh size $\Delta x$. The Courant number is $S=0.3$ in all calculations. In all figures, the red, solid (blue, dashed) curve shows the result without (with) the interface averaging of the dielectric constants. The parameters of the incident wave and the metal slab are the same as in the calculations leading to Fig. 6.Reuse & Permissions

Figure 8

(Color online) Magnetic field snapshots for a plane wave obliquely incident on two layers of materials. The interfaces between the layers and vacuum are indicated by solid horizontal lines. The lower layer is an 80-nm-thick Drude metal with parameters: $ϵ\left(\infty \right)=7.0246$, ${\omega }_D=1.5713\times {10}^{16}\text{rad}/\text{s}$, and ${\Gamma }_D=1.4003\times {10}^{14}\text{rad}/\text{s}$. The upper layer is a 100 nm dielectric with dielectric constant ${ϵ}_r=11.7$. The media below and above the two layers are vacuum. The panels in the right column illustrate the scattering due to a slit of width 200 nm and depth 120 nm in the same layered structure as in the left column. Rows 1, 2, and 3 show snapshots at 1.60, 3.20, 4.80 fs (before a steady state is established); row 4 shows the snapshot at 60.04 fs (after a steady state is established). For all calculations, the incident plane wave enters the TF region from the lower left corner of the TF/SF boundary with incident angle $\theta =65°$. The incidence wavelength is $\lambda =400\text{nm}$ (period $T=1.33\text{fs}$) and the steady-state amplitude of the incident magnetic field is 1 A/m. The mesh size is $\Delta x=2.0\text{nm}$ and the Courant number is $S=0.3$. A log color scale is used in all plots. The thick, dashed (thin, dotted) rectangle indicates the TF/SF (inner PML) boundary. The slabs are extended into the UPML region.Reuse & Permissions

Figure 9

(Color online) Top (input) surface interference signal $I_s\left(x,t\right)$ and surface magnetic field $H_s\left(x,t\right)$ as functions of distance at different instants after excitation by an oblique Gaussian pulse (pulse width 10 fs). In panels (a) through (f), the top portion (green, solid curve) illustrates the signal $I_s\left(x,t\right)$; in the lower portion, the red, dashed curve shows the incident field and the blue, solid curve shows the (pure) scattered field. The inset in panels (c) through (f) illustrates an expanded view of $I_s\left(x,t\right)$ near the left wall of the slit $\left(-2\mu \text{m}<x<0\right)$. For all plots, the units of $I_s$ and $H_s$ are ${10}^{-29}{\left(\text{Coul}/{\text{m}}^2\right)}^4$ and A/m, respectively; $x=0$ corresponds to the right wall of the slit, and $t=0$ corresponds to the instant when the maximum of the incident wave packet (dashed curve) passes $x=0$.Reuse & Permissions